Generic instantaneous force modeling and comprehensive real engagement identification in micro-milling

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Generic instantaneous force modeling and comprehensive real engagement identification in micro-milling

Yu Zhang Methodology; Software; Data Curation; Formal analysis; Validation; Writing- Original draft preparation , Si Li Experimentation; Investigation; Visualization , Kunpeng Zhu Conceptualization; Methodology; Writing- Reviewing and Editing; Resources; Supervision; Project ad PII: DOI: Reference:

S0020-7403(19)32866-8 https://doi.org/10.1016/j.ijmecsci.2020.105504 MS 105504

To appear in:

International Journal of Mechanical Sciences

Received date: Revised date: Accepted date:

4 August 2019 17 January 2020 30 January 2020

Please cite this article as: Yu Zhang Methodology; Software; Data Curation; Formal analysis; Validation; Writing- Or Si Li Experimentation; Investigation; Visualization , Kunpeng Zhu Conceptualization; Methodology; Writing- Review Generic instantaneous force modeling and comprehensive real engagement identification in micro-milling, International Journal of Mechanical Sciences (2020), doi: https://doi.org/10.1016/j.ijmecsci.2020.105504

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Highlights  A new and generic instantaneous force model in micro milling is established, in which the size and tool runout effects are reflected. 

The real cutter-workpiece engagements are comprehensively identified with critical position of actual entry and exit angles explicitly determined.



The proposed force model is developed and can be obtained conveniently and fast by only once calculation of the AUCT with inclusion of tool run out effect.



The theoretical forces obtained from the proposed model are compared and validated with experimental data.

1

Generic instantaneous force modeling and comprehensive real engagement identification in micro-milling

Yu Zhang1,2*, Si Li1, Kunpeng Zhu1,2 1. Institute of Advanced Manufacturing Technology, Hefei Institutes of Physical Science, Chinese Academy of Science, Huihong Building, Changwu Middle Road 801, Changzhou 213164, Jiangsu, China. 2. University of Science and Technology of China, Hefei 230026, China. *Corresponding author: [email protected]

Abstract Cutting force modelling has been one of the most important tools for understanding the dynamics and for process control of micro milling. While current studies have widely investigated the uncut chip thickness and tool runout mechanics, they mostly took account a specific case and a general tool engagement determination is yet to be established. These lead to low adaptability and bring difficulties for process modeling and online control. In this study, a new and generic instantaneous force model is developed, in which the size effect is reflected in the force coefficients, and the tool runout effect is included in the instantaneous uncut chip thickness. In the model the real engagement is comprehensively identified under the tool run out effect, with the actual entry and exit angles at the critical position explicitly determined. The average uncut chip thickness (AUCT), actual cutting depth, center position and geometrical relations are analytically established. The proposed force model can be obtained conveniently and fast by only once calculation of the AUCT with inclusion of tool run out effect. With the main influencing factors considered, the force models are computational efficient, good in adaptability and could conveniently be generalized to 2

universal model. Compared to the experimental measurements, the maximum peak errors of the forces are all less than 0.6%, which validates the efficiency and accuracy of the proposed model. Keywords: Micro-milling; Force modelling; Runout effect; Tool engagement

1 Introduction The cutting force is key to understanding the mechanics of micro milling process, and there have been various force models developed in recent years [1]. Among them, the mechanical force models are most studied and important for their physical significance and their close relationships to the system dynamics. They are generally achieved by modeling the instantaneous uncut chip thickness (UCT) and the force coefficients to establish the model. Because of the size effect and ploughing phenomena, traditional methods on calculating the instantaneous UCT are not applicable to micro milling. The instantaneous UCT model in micro milling is determined by with trochoidal trajectories, tool runout, elastic recovery, and many other mechanisms. The tool path in micro milling is along with trochoidal trajectories due to the large ratio of the workpiece feed to the tool diameter, in which the traditional assumption of circular cutter edge trajectories is no longer applicable. Under this condition, the traditional milling thickness model could be regarded as a special limit case of micro milling, when the ratio of the workpiece feed to the cutter diameter is zero [2]. Bao and Tansel [2] proposed a model for the instantaneous UCT in micro milling, taking into account the actual cutter edge trochoidal trajectories when the micro milling cutter was rotating and feeding simultaneously. It was found that this model had 15% higher prediction accuracy than the traditional milling force model. Similarly, Li et al. [3] presented a method to calculate the instantaneous UCT in micro milling, which was obtained by solving priori equations, and the values of them were approximated with the Taylor coefficients. Other 3

studies also found that the edge radius had a significant effect on the cutting force [4, 5]. An evaluation of four chip thickness models for micro milling were investigated by examining their accuracies on force prediction [6]. The tool runout is an important factor due to high spindle speed in micro milling, which influences the chip formation, tool wear, chatter, and surface roughness. It has significant effect on cutting force variations [7,8], and would eventually lead to higher peak forces and uneven tool wear. Jun et al. [9] found that discontinuous cutting processes were caused by the tool runout, which affected the machining surface accuracy. According to the finite element simulation, Bai et al. [10] showed that the rapid tool wear and breakage were resulted with tool runout, when the micro milling cutter was fitted with a flexible plane parallel to the cutting edge. The tool runout is a key factor affecting cutter edge trajectories in micro milling, which cannot be neglected as in the traditional milling. Many studies were carried on force modeling with inclusion of runout effect, and determined the instantaneous UCTs in the shearing-domain and ploughing-domain regimes [11-13]. By comparing the instantaneous UCTs with different lengths and angles of tool runout, Afazov et al. [14] found that the run out had more significant influences on the instantaneous UCT with lower feed rate. Based on traditional UCT method, Bao and Tansel [15] determined the actual cutter edge trajectories with tool runout consideration, and developed an instantaneous UCT model in micro milling. Li et al. [16] and Chen et al. [17] developed instantaneous UCT models by considering tool runout and the machine tool system vibration, respectively, according to theoretical tool edge trajectories. The accuracies of these improved models are verified by experimental data. Bissacco et al. [18] proposed a micro force model by studying the actual cutter edge trajectories in each milling element with tool runout effect, which showed that the predicted force w as in good agreement with the experimental measurement. More similar studies were found in [19-22] on force modeling with run out consideration. Due to the runout effect, the

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cutting edge is not guaranteed to be involved in cutting workpiece at all positions in the axial cutting depth, and there may be a section of cutting edge that does not cut the workpiece in the tool engagement. Lu et al. [23] proposed a specific prediction model of radial runout of cutting edge, considering of the effects of the extended length of micro-milling cutter and the spindle speed, with which the phenomenon of single-tooth cutting in micro-milling process is analyzed. On this account, the material under multi-flute cutting may not be the part left by the intermediate preceded tooth, but by several previous ones, which were not considered in these studies. Different from macro cutting machining, the edge radius of the tool is comparable in size with the uncut chip thickness in micro machining applications [24]. Liu et al. [25] proved that the edge radius affects material deformation processes with the finite element analysis and orthogonal cutting simulation experiments, which changes the material flow pattern and tool– chip contact length. Lai et al. [26] found the specific quantitative relation between the edge radius and the minimum chip thickness, which is a significant parameter in micro machining. If the instantaneous UCT is less than the minimum chip thickness, the chip will not be formed, and there is no material actually removed as a chip [27]. On this condition, another important factor in micro-milling process modelling that cannot be ignored is elastic recovery of work material, especially at low UCT values. Moges et al. [28] corroborated that the elastic recovery of work material has significant effect on process geometry parameters when UCT is less than the minimum chip thickness. Jun et al. [9] shown that the elastic recovery increases the vibrations of the micro-end mill, especially at low feed rates, and it becomes significant at small axial depth of cuts. Considering the effect of the elastic recovery, Malekian et al. [29] modelled the ploughing forces as proportional to the volume of interference between the tool and the workpiece. Therefore, the edge radius, minimum chip

5

thickness and elastic recovery are representative features of size effects in micro milling, and also have significant influences on micro force modelling. In addition to the instantaneous UCT, the estimation of cutting force coefficients is the key to predicting the micro milling force accurately. The cutting force coefficient is related to the machining conditions closely, while these relations is very complicated and there are no generic approaches to model them. Srinivasa and Shunmugam [30] proposed a methodology to predict cutting force coefficients by considering the effects of edge radius and workpiece properties. The sweep angle caused by helical end mill was included in engagement angle computation which used for further in determining integration limits to calculate cutting forces. Sahoo et al. provided a hybrid modelling approach towards prediction of cutting forces, in which force coefficients were evaluated from FEM simulation [31]. The model of Kao et al. was incorporated with the helix angle of the cutter, and the cutting force coefficients was formulated as a function of average cutting force and cutter geometry such as cutter diameter, number of flutes, cutter’s helix angle [32]. In general, the cutting force coefficients could be obtained by an inverse solution that was satisfied with the force [33]. Based on the literatures, there are two main forms of the cutting force coefficients. The first is force coefficients in constant form [21, 29]. The cutting force coefficients in this form were obtained by a number of experiments, with which the prediction error is minimum in the corresponded machining conditions. However, with different machining conditions, these cutting force coefficients become less effective and more seek to the coefficients in polynomial form [34, 35]. The cutting force coefficients in this form were expressed as polynomial functions, consisting of machining conditions, tool geometric parameters, and instantaneous UCTs at different cutting depths. The specific polynomial coefficients are determined experimentally. The validity range and the prediction accuracy of milling force in this form are higher than the constant one.

6

With many factors that influence the micro milling processes, the instantaneous UCT generally has a complicated calculation (iteration) procedure, which brings difficulties for online process modeling and further control. In this aspect, computational efficiency and applicability are of main concern of the force models. At present, the cutter is usually discretized into a number of elements along the axial direction, and the force of an entire cutter is obtained by numerical integration. An excessive discrete number improves the prediction accuracy, but increases the computational cost and the simulation time. Generally, with the traditional approach, the prediction accuracy and the computational efficiency cannot be compromised. These issues present a barrier to real applications. To decrease the model complexity, Kang and Zheng [36] developed a reduced instantaneous UCT calculating procedure with Fourier coefficients, considering the periodic variation characteristics of the instantaneous UCT. A simplified micro milling force model was proposed in [35] by replacing the instantaneous UCT with the average UCT, by which the efficiency of the micro milling force model was improved. However, the effect of tool runout on the actual cutting entry and exit angles, as well as the varieties of the cutting depth and the equivalent tooth position angle were not analyzed and determined comprehensively. Moreover, as a result of varied machining conditions considered in the model, the algorithm iteration steps and computational cost are high for real applications. Hence, this study aims to establish a new micro milling force model, which would facilitate the modeling and simplify the simulation approach without loss of prediction accuracy. At the same time, the important factors such as run out effect and tool engagement are modeled and their relationships are explicitly derived. The details are discussed in the sections followed.

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2 Modelling of cutting forces in micro milling 2.1 The theoretical cutting force model with runout and size effect To model the cutting force in micro milling, the cutter is divided along the axial direction into a certain number of discrete elements, corresponding to the cutting elements of the workpiece. The entire cutting force on a certain tooth is obtained by numerically integrating the elemental force on all discrete elements. So the tangential, radial, and axial forces (dFt,i (t), dFr,i (t), and dFa,i (t)) at an arbitrary instantaneous tooth position angle θ of the j-th flute is established, as shown in Fig. 1. The ordinal number j = 1, 2, …, M. M is the total teeth number of the cutter. The feed force (Fx,j (t), in the x-axial direction) and the normal force (Fy,j (t), in the y-axial direction) can be obtained with an angle transformation. The cutting forces Fx, Fy, and Fz are the sum of the feed and normal forces of both cutting flute edges.

z

z

n dFt dF a

Milling flute j A

y

αen θ

R

y

y

x

n

ϕj

ap

dz

A

x dFr

A-A

n

x

dFt

dFr dFa

ψ

(a)

(b)

Fig. 1 (a) A diagram of cutting process with dual flutes end milling cutter. (b) The elemental force model of a ball nose end milling cutter. dFt, dFr, and dFt are the tangential, radial, and axial differential forces. θ is the current tooth position angle. αen is the starting angle of the current cutting edge.

In accordance with the position angles of a milling cutter in Fig. 1, the r-axial angle θ is given as:

8

  t , z    j  t    z 

(1)

where  j is the r-axial angle while the z-coordinate, z, of the j-th flute is equal to 0. ψ is the lag angle caused by the helical shape of flutes.  j and ψ are given as:  j  t  =s  2πnt 60   j  1 p   z tan    z  =  R

(2)

where p is the pitch angle with p = 2π/Nt. β is the helix angle of the cutter. The initial angle phase s is equal to  j at the initial sampling time (t = 0), and it’s related to the sampling selection time merely, given as s ∈ [0, p ). At the tooth position angle θ of the j-th flute edge, there is a surface development of the micro milling cutter geometry (Fig. 2). The tangential, radial, and axial elemental forces dFM,j (t, z) are given as:  h  t , z  db  z   dF j  t , z   K    dS  z  

where the coefficient matrix K = [

tc

te

rc

re ].

ac

ae

 j  1, 2,

, Nt 

(3)

The matrix elements Ktc and Krc are the shear

coefficients, and Kte and Kre are the edge force coefficients, in tangential and radial directions respectively. The coefficient matrix K is related to milling parameters and materials of the workpiece and the cutter, which also can be identified by a rapid calibration method with machining tests [37]. The coefficients with milling process parameters are given as:

 cos   n   n   tan c sin  n tan    K tc  s sin n cos 2 n   n   n   tan 2 c sin 2  n   sin   n   n   s  K rc  2 sin n cos  cos n   n   n   tan 2 c sin 2  n   cos   n   n  tan   tan c sin  n K   s ac  sin n cos 2 n   n   n   tan 2 c sin 2  n  9

(4)

where τs is the maximum shear stress of the workpiece material. n is shear angle. αn is nominal rake angle. βn is average friction angle. ηc is chip flow angle. According to Merchant's shear angle theory [38], the shear angle is given as:

n 

π  n n   4 2 2

(5)

The size effect caused by the tool edge radius of the micro-milling cutter is simulated by the equivalent slip surface and equivalent rake angle αt, which can be defined as the angle between the shear plane and the accumulation plane of cutting chip, as shown in Fig. 2 and 3. If UCT is more than the minimum chip thickness hlim, the chip slides along the rake face. Otherwise, the chip will not be formed. Therefore, the nominal rake angle αn is the equivalent rake angle αt [18]:

 min  h, hlim    1 re  

 t  arcsin 

(6)

where the minimum chip thickness hlim of UCT is given as:

hlim  re 1  sin  n 

Tool

Tool h

Elastic recovery

re

(7)

h

re

Workpiece

Workpiece

(a) h < hlim

(b) h > hlim

Removed material

Fig. 2 Chip formation relative to the minimum chip thickness in micro-scale machining. hlim is the minimum chip thickness of the uncut chip thickness, which is mainly affected by the tool edge radius re and the nominal rake angle αn.

The theoretical UCT h in Eq. (3) considering the size effect is then modified accordingly as:

10

  hOS  hlim  hOS h  , z     1  pe  hOS  hOS  hlim 

(8)

where hOS is the theoretical UCT without the size effect, which is obtained with the trochoidal trajectories of cutter geometric characteristics. pe is the elastic recovery rate of specific workpiece material, so pehOS is the height of the elastically recovered material. Considering cutter runout effect in the milling process, the intermediate parameter hOS and related variables are expressed as: hOS  max 0, min  h j 0  h j  hi      h j 0  f r  f z sin    k  i, j  hk  ro cos   z    k  1 p     z    o   z 

(9)

where extra uncut chip thicknesses Δhj and Δhi in Eq. (9) are caused by the tool run-out of the current j-th and previous i-th flutes respectively, and hj0 is the ideal UCT without the tool runout effect. fz is the feed of the workpiece per tooth. ro and γo are tool runout length and angle, respectively. The average friction angle βn in Eq. (4) and (5) is the angle between directions of the average normal pressure N0 and the average friction F0 of the rake face, as shown in Fig. 3. Based on the slip-line theory, the average friction angle βn is given as [39]:

  1  n  arctan  π  1   2 n  2

11

    

(10)

αn

αt Equivalent sliding surface

Tool

ϕj

F0

re h

βn

N0

Workpiece

Feed

Fig. 3 A diagram of the equivalent rake angle αt and average friction angle βn when h < hlim.

According to the Stabler chip flow rule, the relation of the chip flow angle ηc and cutting angle parameters is improved by Morcos [40]:

 cos n   n    tan i   sin  n 

c  arctan 

(11)

where i is the tool inclination angle. According to the maximum shear stress theory, the maximum shear stress τs is taken as a half of the yield limit ??s of the workpiece material, so τs = ??s / 2. Overall, for determining the coefficients with milling process parameters in Eq. (4), a sampling set of orthogonal milling force experiments is carried out, in which the yield limit ??s and tool inclination angle i are obtained and measured. For cylindrical cutters, the radius R is constant to the axial parameter z. Hence, the elemental cutting width and length are given as:

db  z   dz  2 2      R  2 d S z  d r  d z  R           1  1  tan  dz z   z   

(12)

The elemental instantaneous milling force dFj(t,z) = (dFxj, dFyj, dFzj)T (N) in the x-, y- and z-axes directions can be derived by coordinate transferring as:

dF j  t , z   TdFMj  t , z 

12

(13)

where the matrix T transforms the dynamic coordinate system P-tra into the fixed coordinate system O-xyz, given as:   cos   T     sin   0 

 sin   cos  0

0  0 1 

(14)

The cutting force of each flute can be obtained by integrating elemental forces of each layer along the z-axis as:

F j  t    dF j  t , z  z ju

z jd

(15)

where the z-axial upper and lower boundaries of the CWE, zju and zjd, are determined by the raxial angle θ, which are correspond to the entry and exit moments of the CWE respectively. Based on the above formula, the overall model with milling time t can be derived as a single variable of 3D dynamic milling force by summing up the force Fj of each flute. The theoretical milling force model in the x-, y- and z-axes with the discrete element method is given as: Nt

FDEM  t    F j  t 

(16)

j 1

It is noted that this derivation is based on flat end milling, and without loss of generality the model can be modified and extended for ball nose end milling. Under this condition, an axial angel of the CWE is included which leads to a modified UCT representation, and then the other parameters in the model are updated correspondingly. The detail derivations can be found in the Appendix.

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2.2 The theoretical cutting force model To simplify the computation, an actual average uncut chip thickness (AUCT) model ̅ is substituted for the instantaneous uncut chip thickness (UCT) model hj. As in Eq. (2), the AUCT ̅ with milling time is given as:





h    max 0, min h j 0  h j  hi   

(17)

As shown in Eq. (7), and without the tool runout effect, the ideal AUCT in Eq. (17) is derived as: h j 0    

2

1

f z sin  cos 1  cos  2 d  fz  2  1  2  1

(18)

where θ1 and exit angle θ2 are the actual cutting entry angle and exit angle of CWE boundaries, as shown in Fig. 4. The actual cutting depth dav is calculated as:

dav  2  1  r cot 

z

y

θ2 θ1

β

0

θ1

θm

(b)

θm

dav

(a)

θ2 π

θ

(19)

O

Fr,i (θ) Ft,i (θ) h (θ) x

Fig. 4 (a) A surface development with the cutting edge along the circumferential direction; (b) the geometry of an UCT.

There is a linear relation between the runout angle γ and the radial lag angle ψ, as shown in Eq. (7). The tool runout angle γ increases with the increasing of the radial lag angle ψ, and the tooth position angle θ decreases with the increasing of the radial lag angle ψ. At the same cutting depths, the increased angle of the runout angle γ is the same as the decreased angle of 14

the tooth position angle θ, which is equal to the radial angle ψ. Hence, the cutting entry angle θ1 and exit angle θ2 are corresponding to the machining positions where the runout angles are the maximum γ2 and minimum γ1 respectively. The minimum γ1 and maximum γ2 of the cutter runout angle are then given as:

 1   o    2   2   o    1

(20)

Similarly, the average difference of AUCT between two flutes is given as: 2

hk   ro cos    k  1 p  d 1

 rR

sin  2   k  1 p   sin  1   k  1 p 

 2  1

 k  i, j 

(21)

With the above calculation of the actual average UCT ̅ (θ) and the actual cutting depth dav, according to Eq. (3), The tangential, radial, and axial elemental forces dFM,j (t, z) are simplified as:  h j    dFSIM, j    K   d av  1 

 j  1, 2,

, Nt 

(22)

When they are transformed into the fixed coordinate system O-xyz, the force FSIM (θ) describe the feed force and normal force components of the j-th flute edge individually:

FSIM, j    T  m   dFSIM, j  

(23)

The variable θm in the angle transformation matrix T in Eq. (14) is the equivalent position angle of the tangential force Ft,j (θ) and radial force Fr,j (θ), as shown in Fig. 2 (b). It can be considered as the tooth position angle at the actual cutting center approximately, i.e., θm = (θ1 + θ2)/2. Therefore, the simplified instantaneous micro cutting force FSIM (θ) at the tooth position angle θ is the sum of the force FSIM, j (θ) of all cutting flute edges: Nt

Nt

j 1

j 1

FSIM   =  FSIM, j     TSIM  m   dFSIM, j  

15

(24)

In the simplified force model FSIM, there are still two pivotal parameters, the entry angle θ1 and exit angle θ2 of the force model FSIM that are not determined. This will be discussed comprehensively in Section 3. The achieved force FSIM will be compared with the experimental data and the force FLi from the prior method [16] over the tool tooth position angle θ in Section 4, whereby the accuracy of the proposed simplified model is evaluated based on the experiments.

3 The comprehensive actual engagement determination under tool run out effect Based on the above analysis, there are still two unknown parameters to be determined in the force model in the Eq. (24): the actual cutting entry angle θ1 and exit angle θ2 of CWE boundaries. In this section, the actual cutting entry angle θ1 and exit angle θ2 with the tool runout effect are determined and the relationships are derived explicitly.

3.1 Theoretical entry and exit angles A micro cutting process at the bottom of the cutter is shown in Fig. 5. ω is the spindle angular velocity. The starting and ending angles of the current cutting edge are denoted as αen and αex respectively. When it is groove milling, αen = 0°, and αex = 180°. When it is downmilling, αex = 180°; Otherwise, there is αen = 0° with up-milling. According to the geometric parameters of the tool in Fig. 5, and combining the radial lag angle ψ in Eq. (2), when the micro machining is down-milling or up-milling, the cutting range angle α (αex − αen) and the maximum lag angle ψs are given respectively as:

   arccos  R  ae  R     s  ap  R cot   where ae is the working engagement of the cutting edge. ap is the cutting depth.

16

(25)

y θ αen α f

αex

Cutting edge

O

x

Tool

ae

ω

Workpiece

Fig. 5 A micro cutting process at the bottom of a cutter (z = 0). ae is the working engagement of the cutting edge. αen and αex are the starting and ending angles of the current cutting edge, respectively. ω is the spindle angular velocity. f is the feed rate. θ is the current tooth position angle. α is the cutting range angle.

Considering the relation between the cutting range angle α and the maximum lag angle ψs, the micro milling process can be summarized in two machining forms: the form I (α ≥ ψs) and the form II (α < ψs). The cutter’s edge in Fig. 5 can be extended to the circumference. As shown in Table 1, the horizontal axis represents the circumferential expansion length (tooth position angle θ) of the micro cutter, and the vertical axis represents the axial depth (z). The slanted thick line represents the current cutting edge, whose angle is the helix angle β from the horizontal axis. Then the theoretical entry angle θen and exit angle θex in different cutting processes with machining forms I and II are shown in Table 1. According to the cutting edge position, an entire milling process can be divided into three milling stages: incomplete cut-in (A), full milling (B), and incomplete cut-out (C).

Table 1 The theoretical entry angle θen and exit angle θex in an entire milling process with machining forms I and II Milling stage

Milling process Form I

Theoretical entry and exit angles Form II

17

Form I

Form II

z ap

ψs

z ap

α

ψs

α

θen = αen θex = θ

A 0

θen θex

π

θ

0

(αen ≤ θ < αen + ψs)

π

θ

(αen ≤ θ < αex)

α ψs

z ap

θen θex

ψs

z ap

α

θen = θ − ψs θex = θ

B 0

θen

θex π

θ

(αen + ψs ≤ θ < αex) z ap

α

ψs

θen θex

0

π

θ

(αex ≤ θ < αen + ψs) z ap

α

ψs

θen = θ − ψs θex = αex

C 0

θen θex π θ

(αex ≤ θ < αex + ψs) Unmilling

θen = αen θex = αex

θen θex π θ

0

(αex + ψs ≤ θ < αen + ψs) θen = 0 θex = θ

θ < αen or θ ≥ αex + ψs

3.2 Zero-position interval angle Due to the tool runout effect, it cannot be ensured that the involved tooth edge is cutting the workpiece at all positions, even in the cutting range between the theoretical entry angle θen and exit angle θex. There may be a part of tooth edge that is not cutting the workpiece in actual machining. In the traditional discrete method, the judgment, whether the tooth edge is cutting the workpiece or not, depends on the value of the instantaneous UCT at the corresponding elemental cutting edge. When the instantaneous UCT is positive, the current elemental force is analyzed and added to the entire micro cutting force; When the instantaneous UCT is zero, the current tooth edge is not cutting the workpiece, and the current elemental force is zero. However, the traditional method is not available in this simplified cutting force model, for there are not concrete judgments of all elemental cutting edges. The theoretical entry angle θen and exit angle θex cannot be applied directly into this

18

model, so the actual entry angle θ1 and exit angle θ2 are required. The actual entry and exit angles also represent the angle range boundaries of the involved tooth edge, within which the instantaneous UCTs are all positive. Without loss of generality, considering that the micro cutter is long enough and has enough cutting depth, there must be a zero-position for the current cutting edge, where the theoretical instantaneous UCT begins to decrease to zero. The angle between the zero-position and the cutting edge bottom (the cutting depth z = 0, tooth position angle is set as θ, and the runout angle is set as γo) in the plane of the x- and y-axes is denoted as φk. Based on Eq. (17) ~ (21), there is:

 2π  j  k    2πj  f z sin   k   ro cos    o  k   ro cos    o  k   0 M M   

(26)

By solving the Eq. (26), the interval angle φk is calculated as:

  2π  j  k    2πj    o   ro cos  o   f z sin   ro cos  M M     k  arctan    f cos   r sin  2πj     r sin  2π  j  k       o o o o   z M M     

(27)

If φk < 0, the zero-position must be below the micro cutter bottom, which is pointless in determination of the actual entry and exit angles. These undetermined angles are affected only when the zero-position is positive. The greater the angle φk, the larger the instantaneous UCT hk at the same cutting depths. For a micro cutter with multiple edges, the angle between this zero-position and the cutting edge bottom is denoted as the zero-position interval angle φ, which is the smallest one of the values that are greater than zero in all φk. Particularly, the zero-position interval angle φ can be any one of φk when all φk < 0, so it might as well be φ1. Hence, the zero-position interval angle φ is expressed as:

19

 k  k  min      2   1



k  

 max    0  k

 max    0 

 k  1, 2,..., M 

(28)

k

According to the different values of the zero-position interval angle φ, the actual entry angle θ1 and exit angle θ2 with down-milling or up-milling are analyzed in following.

3.3 Actual entry and exit angles under run out effect 3.3.1 The actual entry and exit angles in the milling process with the machining form I There are three stages in the milling process with the machining form I, as shown in Table 1. Depending on the values of the zero-position interval angle φ, the actual entry angle θ1 and exit angle θ2 are analyzed, as shown in Table 2 and 3. When φ < 0, for all k ∈ {1, 2, …, M}, the zero-positions where the theoretical instantaneous UCT is zero are all below the micro cutter bottom (z < 0), as shown in Fig. 6. It means that the judgment whether the tooth edge is cutting the workpiece or not cannot be obtained. The current cutting edge in the theoretical cutting range can only be all or none cutting the workpiece, correspondingly, the actual average UCT is more than or equal to zero. The condition when the actual thickness is zero is easily eliminated in calculations of the micro cutting force. Therefore, whether the tooth edge is all cutting the workpiece or not, the actual entry angle θ1 and exit angle θ2 are the theoretical entry angle θen and exit angle θex respectively, i.e., θ1 = θen and θ2 = θex.

20

φ

(a) z

φ

(b) z

(c) z

φ

ap

ap

ap

θ(γo)

θ(γo) 0

θen θex

(h = 0)

π

θ

0

θen

θex

π θ (h = 0)

0

θen θex

π θ (h = 0)

Fig. 6 Current flute and workpiece in the zero-positions when the zero-position interval angle φ < 0. θen and θex are the theoretical entry and exit angles, respectively. (a) Cutting-in. (b) Full cutting. (c) Cutting-out.

When φ ≥ ψs, the zero-positions are all above the maximum cutting depth (z > ap) for all k, as shown in Fig. 7. Similar to the machining conditions when φ < 0, the actual entry angle θ1 and exit angle θ2 are the theoretical entry angle θen and exit angle θex respectively, i.e., θ1 = θen and θ2 = θex.

(a) z ap

φ

φ

(b) z

(h = 0)

ap

φ

(h = 0)

θ(γo) 0

θen θex

π

θ

(c) z ap θ(γo)

0

θen

θex

π

θ

(h = 0) θ(γo)

0

θen θex π

θ

Fig. 7 Current flute and workpiece in the zero-positions when the zero-position interval angle φ ≥ ψs, where ψs is the maximum lag angle. θen and θex are the theoretical entry and exit angles, respectively. (a) Cutting-in. (b) Full cutting. (c) Cutting-out.

When 0 ≤ φ < ψs, depending on the zero-positions, there are five conditions in the three milling stages with the machining form I. Taking the aforementioned results when φ < 0 and φ ≥ ψs into consideration, the actual entry angle θ1 and exit angle θ2 in an entire milling process with the form I are listed in Table 2.

21

Table 2 The actual entry and exit angles in an entire milling process with the machining form I Milling process with the machining form I Up-milling

Zero-position interval angle φ

Down-milling

φ

θen θex

θex − θen ≤ φ < ψs

π/2 θ

ψs

z ap

θ1 = θen θ2 = θex

φ 0 ≤ φ < ψs – (θex – θen)

h=0 θ (γo)

θenθex π/2 θ

0

φ

z ap

h=0

φ

z ap

h=0

θ (γo) 0

θen θex

π/2 θ

φ

z ap

0 π/2

z ap

θen θex

ψs

θ (γo)

0 π/2 θen θex

π θ

h=0

h=0

θ (γo)

θ (γo)

π θ

φ

0 π/2

z ap

θen θex π/2 θ

θen θex

θ1 = θex − φ θ2 = θex

θ1 = θen θ2 = θex − φ

ψs

ψs – (θex – θen) ≤ φ < ψs

θ1 = θ – φ θ2 = θex

θ1 = θen θ2 = θ − φ

π θ

φ h=0 θ (γo)

0 π/2

0 ≤ φ < θex − θen φ

z ap

h=0 θ (γo) 0

Down-milling

h=0 θ (γo)

0

Up-milling

φ < 0 or φ ≥ ψs

Fig. 6 or Fig. 7 z ap

Actual entry and exit angles

θen θex π θ

3.3.2 The actual entry and exit angles in the milling process with the machining form II There are also three stages in the milling process with the machining form II, as shown in Table 1. When the milling stages are incomplete cut-in (A) and cut-out (C), the actual entry and exit angles with the form II are the same with the form I. Therefore, here are only conditions of the actual entry and exit angles in the full milling stage (B) with form II. When φ < 0 or φ ≥ ψs, the zero-positions are all beyond the cutting range, as same as the conditions with the machining form I in Fig. 6 and 7, as shown in Fig. 8. Similarly, the actual 22

entry angle θ1 and exit angle θ2 are the theoretical entry angle θen and exit angle θex respectively, i.e., θ1 = θen and θ2 = θex.

(a) z

φ

(b)

ap

z

φ (h = 0)

ap

θ (γo) θen θex

0

(h = 0)

θ (γo)

π

θ

0

θen θex

π

θ

Fig. 8 Current flute and workpiece in the zero-positions when αex < θ < αen + ψs, and (a) φ < 0, (b) φ ≥ ψs. θ is the current tooth position angle. αen and αex are the starting and ending angles of the current cutting edge, respectively. φ is the zero-position interval angle. ψs is the maximum lag angle.

When 0 ≤ φ < ψs, depending on the zero-positions, there are two conditions in the full milling stage (B) with the machining form II. Taking the aforementioned results when φ < 0 and φ ≥ ψs into consideration, the actual entry angle θ1 and exit angle θ2 in the full milling stage (B) with the form II are listed in Table 3.

Table 3 The actual entry and exit angles in the full milling stage (B) with the machining form II Full milling stage (B) with the machining form I Up-milling

Zero-position interval angle φ

Down-milling

0 ≤ φ < ψs − (θex − θen)

θ (γo)

z ap

θen θex

π θ

θ1 = θen θ2 = θex

φ

(h = 0)

θ − θen ≤ φ < ψs

θ (γo) 0

Down-milling

φ

(h = 0)

0

Up-milling

φ < 0 or φ ≥ ψs

Fig. 8 z ap

Actual entry and exit angles

θen θex

π θ

23

z ap

0

φ

θen θex

φ

z ap

(h = 0)

(h = 0)

θ (γo)

θ (γo)

π θ

0

θen θex

ψs − (θex − θen) ≤ φ < θ − θen

θ1 = θ − φ θ2 = θex

θ1 = θen θ2 = θ − φ

π θ

3.3.3 The actual entry and exit angles at the critical position In addition to up-milling and down-milling, there is a critical position around where both the two milling methods exist simultaneously in micro machining. For instance, irrespective of the trochoidal trajectory, the boundary of the up-milling and down-milling is at 90° to the y-axis, as shown in Fig. 9.

y Up-milling θ

(θ = π/2) x

O

Down-milling

Fig. 9 A critical position where both up-milling and down-milling exist simultaneously.

When 0 ≤ φ < ψs, and the cutting edge is at the critical position, i.e., π/2 ≤ θ < π/2 + ψs. Since the instantaneous UCT is symmetrical about the feed direction (along the x-axis, 90° to the y-axis), the tooth position angle at zero-position must be between 90° and θ. Depending on the zero-positions, there are eight conditions in in the three milling stages with the machining form I, as shown in Table 4.

24

Table 4 The actual entry and exit angles at the critical position with the machining form I Zero-position interval angle φ

Milling process with the machining form I z ap

φ

h=0

θ (γo) αex π θ

αen π/2

z ap

αenθenπ/2

0

θ (γo) αex π θ

z ap

π + φ − αen ≤ θ < φ + αex

θ1 = αen θ2 = θ − φ

φ h=0

0

Actual entry and exit angles

φ

z ap h=0

0

Tooth position angle θ

αenθenπ/2

0 ≤ φ < θex − θen

θ (γo) αex π θ

π/2 ≤ θ < min{π + φ − αen, αex}

φ

θ1 = π − θ + φ θ2 = θ − φ

h=0 θ (γo)

0

αen π/2

αex π θ

π/2 ≤ θ < min{π + φ − αen, φ + αex}

φ

z ap

h=0 θ (γo) 0

αen θen π/2 αex π θ

z ap

φ

0 ≤ φ < ψs − (θex − θen)

h=0 θ (γo) 0

z ap

αen θen π/2 αex π θ

θ1 = π − θ + φ θ2 = αex

φ

max{π + φ − αen, φ + αex} ≤ θ < π/2 + ψs

h=0 θ (γo) 0

z ap

αen θen π/2 αex π θ

φ

ψs − (θex − θen) ≤ φ < ψs

h=0 θ (γo) 0

φ + αex ≤ θ < π + φ − αen

αen θen π/2 αex π θ

25

π + φ − θen ≤ θ < φ + αex

θ1 = θen θ2 = θ − φ

However, if the cutting depth is not particularly large, the milling time at the critical position is very short. When the sampling frequency is small and the cutting depth is not large, the influence of the critical position is usually not considered.

3.4 Summarization of the actual entry and exit angle determination The actual entry and exit angles in three milling stages are analyzed detailly in Section 3.3. To simplify the determination process of the actual entry and exit angles, an overall analysis is completed and summarized in the following list. a) When the zero-position is beyond the theoretical cutting range, as shown in Table 2 and 3, the actual entry angle θ1 and exit angle θ2 are the theoretical entry angle θen and exit angle θex respectively, i.e., θ1 = θen and θ2 = θex. b) When the zero-position is in the theoretical cutting range in up-milling, as shown in Table 2, the theoretical instantaneous UCT decreases with the increase of the cutting depth. The actual entry angle θ1 = θ − φ, and the actual exit angle θ2 = θex. c) When the zero-position is in the theoretical cutting range in down-milling, as shown in Table 2, the theoretical instantaneous UCT increases with the increase of the cutting depth. The actual entry angle θ1 = θen, and the actual exit angle θ2 = θ − φ. d) When the cutting edge is at the critical position, because the milling time at this position is very short, the influence is not considered when the sampling frequency is small and the cutting depth is not large. This condition can be regarded as down-milling.

4 Experimental validation 4.1 Experimental Setup The traditional discrete method can contribute more accurate prediction results of the micro milling force with a larger sampling frequency. In order to verify the proposed 26

simplified micro milling force model, the results from the traditional discrete method is regarded as the measured micro milling force, which is compared with the prediction force of the simplified model. The traditional discrete element method is used to measure the micro milling force in experiments, which was completed on the five axes precision machining center HSM600U, as shown in Fig. 10 (a). The cutter is a dual flute carbide micro milling cutter, which is commercially available. The diameter is 0.8 mm, and the helix angle is 30°. The work-piece material is pure copper, and was fixed on a three-channel piezoelectric force sensor, as shown in Fig. 10 (b).

(a)

(b)

Fig. 10 Experimental setup: (a) Test platform, machine tool HSM600U; (b) machining process with a micro milling cutter.

A series of tests were conducted on the machine driven by a 22kW spindle drive motor, and the spindle variation is between 18,000 − 30,000 rpm. The machining parameters of 11 tests are listed in Table 5. The milling force was measured with a Kistler 9119AA2 3-channel piezoelectric dynamometer under the work-piece. The measuring range is −4 ~ 4 kN, and the sampling frequency is 50 kHz. The force output was recorded on a Sony digital tape recorder, as shown in Fig. 11 (a) in an entire pass and 11 (b) in a single revolution. For the force calculation of the simplified model, compared to the measuring output force, the micro cutter 27

was divided into 100 discrete cutting elements along the axial direction. The tool runout offset parameters, runout length and angle, are estimated from each sampling test, and the filtering criteria is to minimize Root Mean Square (RMS) errors value of the estimated theoretical force and sampling experimental data, as shown in Table 5 and Fig. 12. It shows that the runout offset positions are mainly concentrated on both sides of the feed direction, and most of the runout offset lengths are less than 2 μm, except for the value in test #1. Based on that, the algorithm for calculation of the simplified micro milling force model is shown in Fig. 13.

Table 5 Machining parameters and estimated tool run-out offset parameters in 11 tests Tool diameter: 0.8mm

Work-piece material: Copper

Tooth number: Dual flutes

Test

#1

#2

#3

#4

#5

#6

#7

#8

#9

#10

#11

n(rpm)

18000

18000

18000

24000

30000

24000

24000

30000

30000

24000

24000

ap (μm)

60

80

100

80

60

60

100

80

100

80

80

v (mm/min)

72

144

216

288

360

192

96

120

240

24

48

r0 (μm)

3.708

2.001

1.001

1.618

0.2128

0.6267

0.7715

1.8754

1.001

1.1256

1.3974

γo (deg.)

259.40

83.16

267.10

50.29

123.70

118.80

91.55

88.05

84.56

76.17

266.40

Fig. 11 Milling force on an entire cutter over milling time, in (a) a whole machining process and (b) a diagram of the experimental force data in a spindle rotation period.

28

Fig. 12 Estimated tool run-out offset parameters with experimental data.

START

END

Input parameters Cutting conditions f, n, ae, ap Tool parameters r, M, ro, γo Force coefficients Ktc,rc,ac,te,re,ae

Calculate FSIM by Eq. (22)

Determine θen and θex

Calculate FSIM,j (θ) by Eq. (21)

FSIM,j (θ) = [0,0,0]T Y

h = 0 ?

N

Calculate

Calculate φ, θ1 and θ2

γ1, γ2, z, h , and θm

Fig. 13 Algorithm for calculation of the micro milling force model with the proposed simplified method.

4.2 Result and discussion Specifically, two of tests with typical mechanical characteristics were selected to show the validation of the model. The micro milling force results in time domains by using the traditional discrete element method and the proposed simplified model are compared in Fig. 14 and 15. It shows that these two proposed theoretical forces are in good agreement with the 29

experimental data, separately. There are two main peaks in a milling period on the theoretical force curves, which correspond to the time when dual flutes are milling the work-piece respectively. Theoretically, the two peaks are equal in an integral milling period. However, as seen in Fig. 14 and 15, two distinct different peaks are constituted primarily due to the tool runout effect. Because of the larger runout offset length in test #1, as shown in Table 5 and Fig. 12, the difference between two adjacent peaks in this test is more significant. As shown in the algorithm of the proposed simplified method in Fig. 13, the milling force coefficients K are determined with Eq. (4) ~ (11) and a sampling set of orthogonal milling force experiments. It’s more convenient than Li’s method [16], and it’s more efficiency with simplified relations of the AUCT in Eq. (17) and elemental forces in Eq. (22). The error between the proposed simplified method and Li’s method is tiny, in which the force differences are less than 0.01 N or 0.03 N, and fluctuate in a small range around 0 N, as shown in Fig. 14 (c) ~ (d) and 15 (c) ~ (d). It indicates that the proposed simplified model can reflect the measuring force features significantly.

30

Fig. 14 Micro milling forces obtained from the experimental data and the proposed model in test #1. (a, b) Forces in the feed and normal directions; (c) The forces and (d) difference between the Li’s discrete method [16] and the proposed model.

31

Fig. 15 Micro milling forces obtained from the experimental data and the proposed model in test #8. (a, b) Forces in the feed and normal directions; (c) The force and (d) difference between the Li’s discrete method [16] and the proposed model.

In order to deeper analysis the error of prediction micro milling forces obtained of two methods, three indicators are used to evaluate results of the proposed simplified model: a) Correlation coefficient ρ:



Cov  FSIM , FLi 

(29)

Var  FSIM  Var  FLi 

where the covariance Cov (FSIM, FLi) = E[(FSIM – E(FSIM))∙( FLi – E(FLi))], and the variance Var (FSIM, Li) = E[(FSIM, Li – E(FSIM, Li))]2. E is the expectation of variables. b) The p-value in the significance test:





p  2 min Pr    1   2 H 

(30)

where the p-value is the probability of observing the given result, or one more extreme, by chance if the null hypothesis is true. Small values of P cast doubt on the validity of the null hypothesis. The bilateral significance level α is selected as 1%. Pr is the conditional probability of correlation coefficient ρ, which follows the Student t-distribution in the statistical hypothesis test H, given as:

H0 :   0   H1 :   0

(31)

where the null hypothesis H0 indicates that there is no linear correlation between the force FSIM and FLi. Hypothesis H1 refers to that there is a linear correlation to a certain extent, which is represented by the correlation coefficient ρ, ρ ∈ [–1, 1]. The null hypothesis H0 is rejected explicitly when p-value > 0.99, which means the correlation of FSIM and FLi is significant.

32

c) Percentage of the peak error e: e   max FSIM   max FLi   max FLi 

(32)

where the evolution parameter max{FSIM} is the micro milling force peak of the proposed simplified model, and max{FLi} is the force peak obtained from the discrete element method, in the feed and normal directions respectively. With 11 tests, three evaluation indicators in Eq. (30) ~ (33) of two force methods are shown in Fig. 16, which all shows that the prediction results of two methods are closely. The difference between them is tiny. As shown in Fig. 16 (a), the correlation coefficients of all tests are more than 0.9995. It indicates high correlation between two methods, which is supported with the significance test p-values in Fig. 16 (b), and that the maximum value of the peak error percentage e is less than 0.6% in Fig. 16 (c).

33

Fig. 16 Statistical analysis with 11 tests. (a) Correlation coefficient ρ; (b) The p-value in the significance test; (d) The peak error percentage e.

The helix angle, cutting depth, and spindle speed are the key parameters that influence the prediction accuracy in the traditional discrete method. Analysis of the tool runout effect is the main work in the proposed simplified model. Taking the test #1 for instance, depending on the significance test p-value and peak error percentage e, the milling force errors of the discrete method and the proposed simplified model with changing of the helix angle, cutting depth, spindle speed, tool runout length and angle are analyzed, as shown in Fig. 17 ~ 21. When one of the key parameters is studied, all the others remain the same, as the machining parameters in test #1. The relation between the significance test p-value and peak error percentage e with the tool helix angle β is shown in Fig. 17. The p-values are all more than 0.992, which shows the two methods is high correlation with different helix angles. The error e (Fx) in the feed direction is decreasing when the helix angle more than 55 deg., as shown in Fig. 17 (b). Regardless of in the feed or normal direction, the maximum value of the peak error percentage e is less than 0.25%. It can be found that the error from the tool helix angle effect on the proposed simplified model can be neglected.

34

Fig. 17 (a) The p-value in the significance test and (b) peak error percentage e with a serial of helix angles β, according to the cutting conditions in test #1.

The relation between the significance test p-value and peak error percentage e with the cutting depth ap is shown in Fig. 18. The p-values are all more than 0.996, which shows the two methods is high correlation with different cutting depths. In the feed direction, the error e (Fx) increases with increasing of the cutting depth. The errors e (Fx) and e (Fy) are all less than 0.25%. Similarly, it shows that the error from the cutting depth effect on the proposed simplified model can be neglected.

Fig. 18 (a) The p-value in the significance test and (b) peak error percentage e with a serial of cutting depths ap, according to the cutting conditions in test #1.

35

The relation between the significance test p-value and peak error percentage e with the spindle speed nt is shown in Fig. 19. The p-values are all more than 0.992, which shows the two methods is high correlation with different spindle speeds. The error e (Fy) decreases with increasing of the spindle speed. Regardless of in the feed or normal direction, the maximum value of the peak error percentage e is less than 0.12%. This tiny percentage shows that the error from the spindle speed on the proposed simplified model can be neglected.

Fig. 19 (a) The p-value in the significance test and (b) peak error percentage e with a serial of spindle speeds nt, according to the cutting conditions in test #1.

The relations between the significance test p-value and peak error percentage e with the tool runout length ro and tool runout angle γo are shown in Fig. 20 and 21. The p-values are all more than 0.992 and 0.995 separately, which shows the two methods is high correlation with the tool runout effect. The error e (Fx) decreases and the error e (Fy) increases with increasing of the tool runout length, and the maximum value is less than 0.2%. It can be found that there is not obvious regularity in changing of the percentage with increasing of the runout angle, and the maximum is less than 0.4%. The percentage suddenly decreases sharply when the tool runout angle is around 90 deg. and 270 deg., in which there may be a critical position where both up-milling and down-milling exist simultaneously. The critical position effect is not considered in the proposed simplified model. Because the maximum percentage 36

is still tiny, the critical position effect on the model can be neglected for simplifying the calculation. Similarly, this tiny percentage shows that the error from the tool runout angle effect on the proposed simplified model can be neglected.

Fig. 20 (a) The p-value in the significance test and (b) peak error percentage e with a serial of runout lengths ro, according to the cutting conditions in test #1.

Fig. 21 (a) The p-value in the significance test and (b) peak error percentage e with a serial of runout angles γo, according to the cutting conditions in test #1.

According to the analysis of four evaluation indicators in two tests, and the relations of the peak error percentage E with the helix angle, cutting depth, spindle speed, the tool runout

37

length and angle, the proposed simplified model has high precision and is consistent with the traditional discrete method. The validity of the proposed simplified model is proved.

5 Conclusions A new micro milling force model with tool runout and size effects is proposed. An actual average uncut chip thickness (AUCT) model is established by taking account of the tool runout effect as well as cutter-workpiece engagement. Depending on the zero-position where the theoretical instantaneous uncut chip thickness begins to decrease to zero, the actual cutting entry and exit angles are explicitly determined. With same cutting parameters, the micro milling force prediction error between the traditional discrete method and the improved model is very small. The maximum peak errors are all less than 0.6% in the experimental validation, which showed that the effectiveness and high accuracy of the proposed model. Further studies could be extended with tool wear inclusion in the model for process monitoring and control purpose. 

Combining analytical and experimental methods, it’s more convenient in determining the milling force coefficients with a sampling set of orthogonal milling force experiments. Considering the size effect of the tool’s flute edge, the UCT h and the minimum chip thickness hlim is related with the milling force coefficients quantificationally.



The relations between the AUCT and elemental forces with the milling force coefficients and processing parameters are simplified, as shown in Eq. (17) and (22). The proposed force model is developed and can be obtained conveniently and fast by only once calculation of the AUCT with inclusion of tool run out effect.



The cutter-workpiece engagements are comprehensively identified with critical position of actual entry and exit angles explicitly determined. There is a zero-position for the current cutting edge, where the theoretical instantaneous UCT begins to decrease to zero.

38

For a micro cutter with multiple edges, the angle between this zero-position and the cutting edge bottom is denoted as the zero-position interval angle φ. Specifically, when the zero-position is in the theoretical cutting range in up/down-milling, the theoretical instantaneous UCT decreases/increases with the increase of the cutting depth. 

The theoretical forces obtained from the proposed model are compared and validated with experimental data and the prior method. The maximum peak errors are all less than 0.6% in the experimental validation. In additions, according to the analysis of the peak error percentage E with the helix angle, cutting depth, spindle speed, the tool runout length and angle, the proposed simplified model has high precision and is consistent with the Li’s traditional discrete method. The validity of the proposed simplified model is proved.

Acknowledgements This project is supported by the Chinese National Key Research and Development Project (Grant No.: 2018YFB1703200), the Chinese Ministry of Science and Technology.

Appendix: The generalization of the model to ball-nose milling The main geometry of a 3-flute ball-nose end milling cutter with a constant helical lead is shown in Fig. 22 (a). For a ball nose end milling cutter, one of significant differences from the proposed model is calculation of UCT h, which is affected with the mutative r-axial radius, as shown in Fig. 22 (b). The hemispherical center of the cutter is Q, and the Cartesian coordinate system O-xyz is established with a point O, the hemisphere vertex, as its origin. The point P is on the cutter edge of the j-th flute. The ball-nose radius ̅̅̅̅ = R. The zcoordinate of point P is z (z > 0). The spindle speed nt is set in the clockwise direction. Hence, the radius parameter r at point P is R∙sinκ, where κ is the a-axial angle of the cutterworkpiece engagement (CWE), which is expressed as:

39

arccos 1  z R    π 2  z  R 

 

 z  R

(33)

z nt

dz

Q κR z

P dFa O

r dFt dFr

y

x

Fig. 22 (a) Cutting process with a dual flutes ball nose end milling cutter. (b) The elemental force model of a ball nose end milling cutter. dFt, dFr, and dFt are the tangential, radial, and axial differential forces.

The neighborhood at the point P in the z-axis direction is chosen as a milling element, whose length is dz (mm) on the workpiece correspondingly. The dynamic Cartesian coordinate system P-tra is established with the point P as its origin, and the t-, r- and a-axes are paralleled to the directions of instantaneous tangential, radial, and axial milling forces respectively. The lag angle ψ in Eq. (2) is modified as:

  z=

z tan  r

(34)

There are some main process parameters in the elemental forces model in Eq. (3) and Eq. (13). Correspondingly, the elemental cutting width and length in Eq. (12) are modified as:

db  z   dz  csc   2 2      r  d S z  d r  d z  r         1  z   z   And the transforms matrix T in Eq. (14) is developed as:

40

(35)

  cos   T   sin   0 

 sin  sin   sin  cos  cos 

 cos  sin     cos  cos    sin  

(36)

Because of the variable radius r, parameters of the UCT h in Eq. (9) is improved as: h j 0  f r sin   f z sin  j sin    hk  ro sin  cos   z    k  1 p      z    o   z 

 k  i, j 

(37)

Based on the above improved formulas, the theoretical force model in Eq. (16) is reestablished for applying in ball-nose micro end milling, in which relevant parameters are developed with corresponding geometric characteristics.

Author Statement Yu Zhang: Methodology, Software, Data Curation, Formal analysis, Validation, WritingOriginal draft preparation. Si Li: Experimentation, Investigation, Visualization. Kunpeng Zhu: Conceptualization, Methodology, Writing- Reviewing and Editing, Resources, Supervision, Project administration, Funding acquisition.

Declaration of Interest Statement This is claimed that there is no confict of interest of the authors, Yu Zhang, Si Li, Kunpeng Zhu, for the submision entitled “Generic instantaneous force modeling and comprehensive real engagement identification in micro-milling” for possible publication in the journal of International Journal of Mechanical Sciences.

41

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GRAPHICAL ABSTRACT

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